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Question:

Let n(>1) be a composite integer such that $\sqrt{n}$ is not an integer.

Consider the following statements:

A: n has a perfect integer - valued divisor which is greater than 1 and less than $\sqrt{n}$ .

B: n has a perfect integer- valued divisor which is greater than $\sqrt{n}$ but less than n.

Then,

Explanation:

Consider a number n = 10, then $\sqrt{n} \approx 36$ .

A: We have a divisor 2 which is greater than 1 and less than 3.16.

B: We have a divisor 5 which is greater than 3.16 but less than 10.

∴ Both statements A and B are true.

Also, as a rule, any composite number which is not a perfect square has at least one factor less than $\sqrt{n}$ and another factor more than n, such that their product is n.

∴ Both statements are true.

Hence, option (d).

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